There is pus at the root of a crowned tooth. Can it be treated through a hole instead of removing the crown?

Yes, it is possible to drill through the crown and perform root canal treatment, and the hole can then be filled so you keep using the crown. However, 1) the view through the hole is narrow, so accuracy can suffer, 2) it is hard to check for decay or cracks under the crown, and 3) if the crown is damaged during treatment it has to be remade anyway. For that reason the standard approach is to remove the crown, treat the tooth, and fit a new one. If the affected tooth is part of a bridge spanning several teeth, drilling through offers a large cost advantage and may be worth trying despite those drawbacks. It has to be judged from the condition of the crown, whether decay is present, and the extent of the inflammation, so please have a diagnosis and discuss it with us.